Exotic Differential Operators on Complex Minimal Nilpotent Orbits

Let O be the minimal nilpotent adjoint orbit in a classical complex semisimple Lie algebra g. O is a smooth quasi-affine variety stable under the Euler dilation action C^* on g. The algebra of differential operators on O is D(O)=D(Cl(O)) where the closure Cl(O) is a singular cone in g. See and for some results on the geometry and quantization of O. We construct an explicit subspace A_-1⊂ D(O) of commuting differential operators which are Euler homogeneous of degree -1. The space A_-1 is finite-dimensional, g-stable and carries the adjoint representation. A_-1 consists of (for g ≠ sp(2n,C)) non-obvious order 4 differential operators obtained by quantizing symbols we obtained previously. These operators are "exotic" in that there is (apparently) no geometric or algebraic theory which explains them. The algebra generated by A_-1 is a maximal commutative subalgebra A of D(X). We find a G-equivariant algebra isomorphism R(O) to A, f↦ D_f, such that the formula (f|g)=(constant term ofD_g̅ f) defines a positive-definite Hermitian inner product on R(O). We will use these operators D_f to quantize O in a subsequent paper.